Hi Randy. I'm coming to this party a little late but I'm another happy camper with a 2011 4WD Suburban 1500 LTZ which I bought last year with 231,000 on the odometer and lots of maintenance records. My 17' Winnebago weighs in between 4000 and 4200 depending on the length of trip and my wife's packing, which is a bit lighter than your target but I can say I don't know it's back there unless I'm in a real crosswind. That said, I've put some serious effort into bulletproofing it, even with those receipts for history. As others have mentioned, that started with ditching the air shock system and installing Bilstein 5100s all around. I also replaced the 1000 lb rear springs with 1250 lb Moogs (pretty sure they are the Z-71 spec springs) and set the ride height in the front to match the increased height in the rear from the stiffer springs. It is a wife-approved upgrade and handles the trailer very well.
Up front, I replaced a leaking radiator with a 3" Cold Case aluminum radiator and upgraded the stock HD transmission oil cooler with a Tru-Cool 40k, which is a monster, and finally, a bigger battery when the one that came with the truck gave up the ghost. It goes without saying we changed out all the fluids - brake, oil, full transmission service, T-cases, and diffs. New plugs and wires. I'm running 32-ish inch tires on 18" wheels and am starting to wonder if maybe 3.73s might not be a good idea. I'm also going to keep an eye on my engine temps on a long trip this summer and if I don't like what I see I may switch out the Tru-Cool for a Derale with fans that can be located somewhere other than in front of the radiator. But I'm hoping the oversize two-row Cold Case keeps things under control. When I upgrade to lithium batteries in the trailer, I'll get a big Mechman 250 - 350 amp alternator, but that's a year or so down the road. There's a few other mods that pertain to not getting caught out in the wilderness without gas or air or such, but they don't impact towing ability (you can see them in my signature).
As far as a hitch system, our dealer installed a Husky WDH with 800-1200 lb bars that take care of both equalizing the ride height and controlling sway. I've used the Haul-Master system above - years ago my family pulled a 30 ft. Holiday Rambler with a 3/4 ton International Travelall all over the west - and that's all we had back in those days. My personal feeling is the new designs which incorporate anti-sway with the weight equalizing function are superior but they both work.
We're headed to Michigan's Upper Peninsula for three weeks this summer and hope to make a three+ week circle through Big Bend National Park, Carlsbad Caverns and Roswell in New Mexico, and back to Palo Duro Canyon from Chattanooga, hopefully as soon as this winter. We'll be packing 10 extra gallons of gas, 10 extra gallons of drinking water, plenty of tools, a big puncture repair kit for the tires and heading out without fear. No freeways at all - 2 lanes only. At 114K miles, yours is just getting broken in. Do the maintenance and the upgrades you think appropriate and go explore!
Bob